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Algebraic Cycles and Stark-Heegner Points

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Algebraic Cycles and Stark-Heegner Points Algebraic cycles and Stark-Heegner points Henri Darmon Victor Rotger July 2, 2011 2 Contents 1 The conjecture of Birch and Swinnerton-Dyer 7 1.1 Prelude: units in number fields . 7 1.2 Rational points on elliptic curves . 10 2 Modular forms and analytic continuation 15 2.1 Quaternionic Shimura varieties . 15 2.2 Modular forms on quaternion algebras . 17 2.3 Modularity over totally real fields . 19 3 Heegner points 23 3.1 Definition and construction . 23 3.2 Numerical calculation . 27 4 Algebraic cycles and de Rham cohomology 33 4.1 Algebraic cycles . 33 4.2 de Rham cohomology . 35 4.3 Cycle classes and Abel-Jacobi maps . 40 5 Chow-Heegner points 43 5.1 Generalities . 43 5.2 Generalised Heegner cycles . 45 6 Motives and L-functions 51 6.1 Motives . 51 6.2 L-functions . 56 6.3 The Birch and Swinnerton-Dyer conjecture, revisited . 60 7 Triple product of Kuga-Sato varieties 65 7.1 Diagonal cycles . 65 7.2 Triple Chow-Heegner points . 68 8 p-adic L-functions 71 8.1 p-adic families of motives . 71 3 4 CONTENTS 8.2 p-adic L-functions . 76 9 Two p-adic Gross-Zagier formulae 83 10 Stark-Heegner points and Artin representations 87 Introduction Stark-Heegner points are canonical global points on elliptic curves or modular abelian vari- eties which admit{often conjecturally{explicit analytic constructions as well as direct rela- tions to L-series, both complex and p-adic. The ultimate aim of these notes is to present a new approach to this theory based on algebraic cycles, Rankin triple product L-functions, and p-adic families of modular forms. It is hoped that the reader could eventally use this text to get an understanding of the authors' strategy and of how it relates to the influential point of view on Euler systems and p-adic L-functions pioneered by Kato and Perrin-Riou. In their current, very preliminary incarnation, the notes are meant to serve as a com- panion to the authors' lectures at the 2011 Arizona Winter School, and their scope is more limited. In spite of this, they still include a broader range of topics than could adequately be covered in the lectures, where the authors strived to present a narrower selection of topics, hopefully in somewhat greater depth. In particular, the AWS lectures only cover the material in Chapters 1 to 7, in which no p-adic methods are involved. The topics covered in the last chapters are only included because of the (circumstantial, indirect) support they provide for the authors' belief that the study of diagonal cycles on the product of three Kuga-Sato varieties could eventually shed new light on some of the more mysterious aspects of the Stark-Heegner point construction. The survey concludes with a bibliography giving a complete1 list of the articles about Stark-Heegner points that have appeared in the literature until now. These articles are as- signed labels like [LRV] or [Gre09], corresponding to the authors' initials followed eventually by a year of publication for items having already appeared. To limit the length of this bibli- ography, each chapter also contains a shorter list of more local references that are germane but not directly related to Stark-Heegner points, numbered consecutively within the chapter in which they are cited. Acknowledgements. It is a pleasure to thank Taylor Dupuy, Luis Garc´ıa, Xavier Guitart, Ariel Pacetti, Kartik Prasanna and Bjorn Poonen for their comments, suggestions and corrections. 1at the time of writing, and to the best of our incomplete and fallible understanding! The authors apologise in advance to anyone who feels their contributions were omitted, underplayed, or misrepresented, and welcome suggestions for clarifications or additions to the bibliography. 5 6 CONTENTS Chapter 1 The conjecture of Birch and Swinnerton-Dyer 1.1 Prelude: units in number fields The problem of calculating the unit groups of real quadratic fields, which essentially amounts to solving Pell's equation x2 − Dy2 = 1; (1.1) is among the most ancient Diophantine problems, with systematic approaches to it dating back to the 7th century Indian mathematician and astronomer Brahmagupta. The early algorithms for solving Pell's equation can all be reduced to the method of continued frac- tions in various guises (or, as Fermat saw it, to his general method of descent). Of great importance for this survey are the following two more sophisticated approaches which grew out of Dirichlet's seminal work on primes in arithmetic progressions: × 1. Circular units. Let χ :(pZ=DZ) −! ± 1 be an even primitive Dirichlet character of conductor D, and let F = Q( D) be the real quadratic field attached to χ by class field theory. In [1.1], Dirichlet obtains the striking identity D h Y 2πim=D χ(m) "F = (1 − e ) ; (1.2) m=1 where "F is a fundamental unit of F of norm one, the integer h ≥ 1 is the class number of x P1 n F and e = n=0 x =n! is the usual exponential function. Implicit in this identity is the containment F ⊂ Q(e2πi=D), a special case of the Kronecker-Weber theorem that is central to Gauss's fourth proof of the law of quadratic reciprocity. The expressions in the right hand side of (1.2) are the simplest examples of circular units, which in general give rise to a systematic collection of units in abelian extensions of Q and are of fundamental importance in the classical theory of cyclotomic fields. 2. The class number formula. The analytic class number formula for the Dirichlet L-series L(s; χ) (combined with the functional equation relating its values at s and 1 − s) 7 8 CHAPTER 1. THE CONJECTURE OF BIRCH AND SWINNERTON-DYER asserts that p 0 2h log "F = L (0; χ) = 2 DL(1; χ): (1.3) One can therefore recover "F (or at least its 2h-th power) by exponentiating the special value L0(0; χ). Both the approaches based on circular units and on special values of L-series lead to poor algorithms for solving Pell's equation, the simpler continued fraction method being far superior in practical terms; as Dirichlet himself writes in [1.1], \Il est sans doute inutile d'ajouter que le mode de solution que nous allons indiquer, est beaucoup moins propre au calcul num´eriqueque celui qui d´erive de l'emploi des fractions continues et que cette nouvelle mani`erede r´esoudre l'´equation t2 − pu2 = 1, ne doit ^etreenvisag´eeque sous le rapport th´eorique comme un rapprochement entre deux branches de la science des nombres." The \two branches" of number theory which Dirichlet alludes to can be broadly construed as arithmetic{the theory of integers and discrete quantities{ and analysis{concerned with limits and continuous quantities. These branches have become closely intertwined since the 19th Century. It is now abundantly clear that both (1.2) and (1.3) epitomize a vigorous theme in modern number theory: the explicit construction, by analytic means, of solutions to naturally occurring Diophantine equations. The goal of these notes is to give a (biased, and incomplete) survey of a few aspects of this classical theme, with special emphasis on the case where unit groups of real quadratic fields are replaced by Mordell-Weil groups of elliptic curves over various global fields. Even though (1.2) and (1.3) are essentially equivalent (cf., exercise 1.1 below) it is nonetheless worthwhile to draw a clear distinction between the two. This is partly be- cause the theory of circular units underlying (1.2) is notoriously difficult to extend to other settings. Only when the ground field Q is replaced by a quadratic imaginary field K are circular units known to admit satisfactory analogues: the so-called elliptic units obtained by evaluating the Dedekind eta-function at arguments in K. Extending (1.2) to abelian extensions of more general number fields is one of the most tantalising questions in number theory, with close ties to Hilbert's twelfth problem and explicit class field theory. On the other hand, equation (1.3) is the simplest non-trivial instance of the analytic class number formula which admits a well-known formulation for general number fields. If K is such a number field with class number hK , discriminant DK and ring of integers OK , and r1 and 2r2 denote the number of distinct real and complex embeddings of K respectively, then × the unit group OK is a finitely generated group of the form × r OK ' (Z=wK Z) × Z ; where r = r1 + r2 − 1; (1.4) by Dirichlet's unit theorem. The integer wK appearing in this equation is the number of roots of unity in K. The general class number formula asserts that r1 r2 2 (2π) hK RK ress=1 ζK (s) = p ; (1.5) wK jDK j 1.1. PRELUDE: UNITS IN NUMBER FIELDS 9 where ζK (s) is the Dedekind zeta-function of K, and RK is the regulator of K obtained by choosing a system "1;:::"r of generators for the units of K modulo torsion and taking the determinant of an r × r matrix of logarithms of these units relative to any set v1; : : : ; vr of distinct embeddings of K into R or C: RK := det (log jvj("i)j)1≤i;j≤r : (1.6) It is sometimes more suggestive to rephrase (1.5) in terms of the behaviour of ζK (s) at s = 0, by applying the functional equation relating its values at s and 1 − s. The class number formula then becomes −r hK RK ords=0 ζK (s) = r; and lim s ζK (s) = : (1.7) s!0 × #OK When r = 1 and K has a real embedding, (1.7) leads to an analytic expression for a unit of K, 0 essentially by exponentiating ζK (0).
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